Equi-Affine Riemannian Metrics in A³

• Equi-affine Riemannian metrics are invariant constructions that use higher-order derivatives and mixed determinants to define arc lengths and first fundamental forms in A³.
• They employ methods like the affine first fundamental form for surfaces and a sixth-root determinant for curves, ensuring invariance under volume-preserving transformations.
• A commensurability criterion algebraically relates intrinsic and induced arc lengths, offering critical insights into the geometry of nondegenerate curves and surfaces.

Equi-affine Riemannian metrics provide a rigorous framework for constructing invariant metric geometries in equiaffine 3-space ($A^3$), despite the absence of any inner product preserved by the full equiaffine symmetry group. For nondegenerate curves and surfaces, canonical notions of “arc length” and “first fundamental form” admit fully equiaffine invariance. These constructions assign, to each nondegenerate surface, an equi-affine first fundamental form, and to each nondegenerate curve, an equi-affine arc length element—both derived from higher-order derivatives and mixed determinants of local parametrizations. When a curve lies on a surface, two generally distinct arc length notions arise; a commensurability condition algebraically characterizes when they coincide (Clelland et al., 2012).

1. Absence of Invariant Inner Product in Equiaffine Geometry

In classical Euclidean geometry, all metric notions—including arc length for curves and the first fundamental form for surfaces—derive from the Euclidean inner product, which is preserved under translations, rotations, and reflections. In contrast, equiaffine geometry, defined by the group of affine volume-preserving transformations, does not admit an invariant inner product on tangent vectors. Consequently, metric-like structures in equiaffine $A^3$ must arise through alternative constructions based on volume invariants and mixed determinants, rather than direct analogues of the Euclidean metric.

2. Affine First Fundamental Form for Nondegenerate Surfaces

Given a local parameterization $X: U \subset \mathbb{R}^2 \to A^3$ of a regular nondegenerate surface $\Sigma$, mixed determinants of tangent and second derivative vectors define the invariants: $$\ell = \det[X_u, X_v, X_{uu}], \quad m = \det[X_u, X_v, X_{uv}], \quad n = \det[X_u, X_v, X_{vv}].$$ The quadratic form

$Q(u, v) = \ell \, du^2 + 2m \, du\,dv + n \, dv^2$

is invariant under the affine volume-preserving group up to a fourth-power factor under reparametrizations. The associated affine first fundamental form is

$\boxed{ \Iaff = \lvert \ell n - m^2 \rvert^{-1/4} (\ell\, du^2 + 2m\, du\,dv + n\, dv^2) }$

which yields a globally well-defined, fully equiaffine-invariant quadratic form on $\Sigma$. For comparison, in the presence of the second fundamental form $\II_{\text{Euc}}$ and Gauss curvature $K$ (from the Euclidean structure), the relation

$\Iaff = |K|^{-1/4} \, \II_{\text{Euc}}$

provides a Euclidean-invariant expression for $\Iaff$ (Clelland et al., 2012).

3. Equi-affine Arc Length for Nondegenerate Curves

For a nondegenerate space curve $\alpha: I \to A^3$, the equi-affine arc length element is determined by the sixth root of the determinant of the $3 \times 3$ frame $\big[\alpha', \alpha'', \alpha'''\big]$: $$\boxed{ ds_\alpha = \sqrt[6]{\det[\alpha'(t),\,\alpha''(t),\,\alpha'''(t)]} \; dt }$$ The total equi-affine arc length from $t_0$ to $t$ is

$$\boxed{ s_\alpha(t) = \int_{t_0}^{t} \sqrt[6]{\det[\alpha'(\tau),\,\alpha''(\tau),\,\alpha'''(\tau)]} \; d\tau }$$

This construction ensures equiaffine invariance and depends crucially on the nondegeneracy condition that $\alpha', \alpha'', \alpha'''$ are linearly independent everywhere along the curve.

4. Induced and Intrinsic Arc Lengths: Two Metrics for Curves in Surfaces

When a nondegenerate curve $\alpha$ lies on a nondegenerate surface $\Sigma$, two natural arc length functions arise. The first, $s_\alpha$, is intrinsic and constructed as above. The second, $s_\Sigma$, is induced by restricting the affine first fundamental form $\Iaff$ to the tangent vector of the curve: $\boxed{ s_\Sigma(t) = \int_{t_0}^{t} \sqrt{\Iaff(\alpha'(\tau))} \; d\tau }$ Thus, for a curve $\alpha \subset \Sigma \subset A^3$, $s_\alpha$ and $s_\Sigma$ represent, a priori, different notions of arc length, depending on third- and second-derivative data, respectively.

5. The Commensurability Criterion and Algebraic Characterization

A nondegenerate curve $\alpha$ in a surface $\Sigma$ is termed commensurate if and only if $s_\alpha(t) \equiv s_\Sigma(t)$ up to a constant shift in base point. Clelland et al. establish that this holds if and only if, for every $t$,

$\det[\alpha', \alpha'', \alpha'''] = [\Iaff(\alpha')]^3.$

Equivalently, expressing these invariants in terms of classical Euclidean invariants—curvature $\kappa$, torsion $\tau$ of $\alpha$, Gauss curvature $K$ of $\Sigma$, and normal curvature $k_n$ of $\Sigma$ in the direction $\alpha'$—yields

$\det[\alpha', \alpha'', \alpha'''] = \|\alpha'\|^6 \kappa^2 \tau, \qquad \Iaff(\alpha') = \|\alpha'\|^2 (|K|^{-1/4} k_n),$

and thus commensurability is characterized by

$\boxed{ \kappa^2 \tau = (|K|^{-1/4} k_n)^3 }$

This commensurability criterion gives a clean algebraic characterization for the coincidence of the two arc length functions; it is equivalent to the vanishing of the “two-arc-length discrepancy” (Clelland et al., 2012).

6. Exemplification: Cases of Commensurability and Discrepancy

Consider the unit 2-sphere $S^2 \subset \mathbb{R}^3$ with the standard parametrization $X(u, v) = (\cos u\cos v,\, \sin u\cos v,\, \sin v)$, so $K \equiv 1$ and $\Iaff = E\, du^2 + G\, dv^2 = \cos^2 v\, du^2 + dv^2$. The spherical helix $\alpha(t) = X(8t, t)$ yields

$$s_\Sigma'(t) = \sqrt{1 + 64 \cos^2 t}, \qquad s_\alpha'(t) = \sqrt[6]{48 \cos t (43 + 672 \cos^2 t)}$$

demonstrating manifestly different arc length densities; thus, the helix is not commensurate. The condition $\kappa^2 \tau \neq (k_n)^3$ likewise fails.

In contrast, any curve on $S^2$ satisfying $\kappa^2 \tau = 1$ (with $K \equiv 1$, $k_n \equiv 1$) achieves commensurability, so both metric integrals coincide and both reduce to the Euclidean arc parameter $s$. The explicit parametrizations for such curves generally require solving a third-order ODE; a one-parameter family of solutions exists, all exhibiting exact agreement between the two arc length measures.

7. Summary and Significance in Equiaffine Metric Geometry

Equi-affine $A^3$ lacks an invariant inner product but admits natural, functorially defined, fully equiaffine-invariant metric objects: the affine arc length for nondegenerate curves (third derivative data) and the affine first fundamental form for nondegenerate surfaces (second derivative data). For embedded curves, the commensurability criterion provides a definitive algebraic test for the coincidence of these metrics, illuminating the structure of two arc length notions and enabling precise classification of their agreement (Clelland et al., 2012). This framework highlights foundational differences from classical metric geometry and reveals deep connections between volume forms, curvature invariants, and affine symmetries in differential geometry.